How can I cut a circular hole into this curvy shape?
Actually it looks like the "crown" is a distinct part:



So I would create 2 different objects, and for the main part I would do it this way:


Create your basic shape from a subdivided cube:





Use the knife tool to cut (K to activate, Z to cut through)





Press V to separate, delete the useless part.





Mirror the shape, merge some vertices with alt M to keep a clean topology:





Give your object a Subdivision Surface modifier, extrude the hole, add some edge loops to sharp the edges:





Alternative topology (no triangle):




You could also use a Shrinkwrap modifier to get a different topology: